J. of Inequal. & Appl., 1997, Vol. 1, pp. 223-238 Reprints available directly from the publisher Photocopying permitted by license only (C) 1997 OPA (Overseas Publishers Association) Amsterdam B.V. Published in The Netherlands under license by Gordon and Breach Science Publishers Printed in Malaysia On Weighted Hardy Inequalities on Semiaxis for Functions Vanishing at the Endpoints MASHA NASYROVAa and VLADIMIR STEPANOVb,*,t Khabarovsk State University, Department of Applied Mathematics, Tichookeanskaya, 136, Khabarovsk, 680035, Russia b Computer Center of the Far-Eastern Branch of the Russian Academy of Sciences, Shelest 118-205, Khabarovsk, 680042, Russia E-mail: 1600 @ as.khabarovsk.su (Received 15 July 1996) We study the weighted Hardy inequalities on the semiaxis of the form for functions vanishing at the endpoints together with derivatives up to the order k k 2 is completely characterized. 1. The case Keywords: Weighted Hardy’s inequality. 1991 Mathematics Subject Classification: Primary 26D 10; Secondary 34B05, 46N20. 1 INTRODUCTION In the theory of the weighted Hardy inequalities for higher-order derivatives the problem, when functions vanish together with derivatives at the endpoints, is still undecided. The case k 1 has been solved by P. Gurka 1 and activity in this area has increased in recent years mostly due to the efforts of A. Kufner, *Author for correspondence. ?The research work of the second author was partially supported by INTAS project grant 94-881 and EPSRC grant GR-K76344 of Great Britain. 223 M. NASYROVA and V. STEPANOV 224 who contributed to various other cases by himself and with coauthors as well [2-6]. We study the problem on the semiaxis, which is different in some aspects from the case of a finite interval [3,9] since the inequality (1), k > 1, becomes non-trivial with only one condition F (cx) 0. We give a complete solution for the ase k 2 and hope that our approach can be extended to the general situation on the semiaxis. The inequality (1), k > 1, with the boundary conditions O, F (j) (oc) F (i) (0) for some 0 < i, j < k 0 (2) 1 becomes non-trivial, if the right-hand side of (1) for all functions, having absolutely continuous derivative F (k- 1) (x) and satisfying (2), is a norm in a corresponding weighted space. We suppose throughout the paper, that u (x) as well as v(x) and Iv(x)1-1 be locally square integrable; the last is necessary and sufficient for the condition, that any function F satisfying F(k)v < cxz, can be represented by a Riemann-Liouville integral. Let o/=(o/0, o/1 k- 1; lal-0/k-l), o/j-0, 1; j=0, 1 112 Put F (’) (0) 0 F () ((x) . O<j<k-1 F (j) (0) F (i) (cxz) 0 for all j, when o/j 0 0 for all i, when fli 1, 1 and I’2 A C (O/ 2 F() (0) F () (oc) 0}. For a finite interval and k > 1 the boundary value problem F () (x) 0, F (a) (0) 0, F () (a) (3) 0 has a non-trivial solution, if 1 < lal / I1 _< k 1. Hence, it might have only the trivial solution, only if lal / I1 >_ k. On the other hand, (3) has only the trivial solution, if la1-4- I1 >_ 2Ck 1) while if lal / I1 >_ 2k, then functions in functions belonging to (a, ) can be approximated by the norm of ACe (o/, ). Thus, on a finite interval we have AC Nk C 22k- -t (2k)! 2(k !)2 WEIGHTED HARDY INEQUALITIES 225 ACke(ot, fl), k IIFIIAc(,,) is not a < I1 + I1 2k; however, for some of them norm. The last number is given by rather more complicated formulae. The situation on (0, cx) is slightly different and affected by the following observation. Let (1) holds and fl(1) and fl2) be multiindices such, that min{i" jl) 1} 1}. Then min{i" fl(2)). It implies an heuristic principle, that the ACk2(ot, fl(1)) characterization problems of (1) for functions with the boundary couples (or, fl(1)) and (or, fl(2)) are equivalent. Similar effects take place on (-x, 0) or (-x, x). For simplicity we restrict ourselves to case of L2-norm, however the methods of the present paper in conjunction with the results 12-14] provide the Lp-Lq case as well. Throughout the paper uncertainties of the form 0.c, 0/0, x/x are taken equal to zero, the inequality A << B means A < cB with an absolute constant B is c, perhaps, different in different places, however the relationship A interpreted as A << B << A or A cB. XE denotes the characteristic function of a set E. spaces fl2) AC(ot, 2 REMARKS ON GENERAL CASE If k > l, then the following characterization are known. (1) (2) (3) (4) (5) k 1. Io1 / I/1 1 [7], I1 / Itl 2 [1]. k > 1. Icl k, I1 0 or I1 0, I1 k [10]. k > 1. I1 / I/1 k, max{j otj 1} + 1 min{i fli 1} [2]. k > 1. Finite interval, Icl / I1 k [9]. 1 and the k > 1. Finite interval, Icl / I1 k / 1, Ok/ /k/l remaining parts of multiindices satisfy the P61ya condition [4-5]. _ Below (Theorem 1.2) we supplement this list by the following (6) k > 1. Infinite interval (a, x), 1 < Io1 / I1 < k, I1 >_ 1, j 0, 1 Io1, I1 / 1 min{i /i 1}. Let (a, b) 1, (-cx, o), a < b and k > 1. We need the following notations. Le, v Ikf (x) j L2,v,(a,b)-- lk,(a,b)f (x) [if" II/vll 1 1-’(k) (x < y)-I f(y) dy, x e (a, b), M. NASYROVA and V. STEPANOV 226 Jkg(X) e,o ek,o;(a,b),u v2 2 e k,1 e 2k,l’(a,b),u,v n, B, I"(k) (y (X sup x)k-l g(y) dy, x (a, b), t)2(k-1)lu(x)12 dx Iv1-2, a<t<b lul 2 sup x)2(k-1)lv(x)1-2dx (t a<t<b 2 Bk,O;(a,b),u,v 0 if b Jk,(a,b)g(x) sup a<t<b Bk,2 1" (a,b),u,v fa b 12f Iv1-2 x )2(k-l) lu(x) dx (t lu[ 2 sup t)2(l-l)lv(x)l-2 dx (x a<t<b a a;(a,b),u,v max(a2,0, a2k,1), S B;(a,b),u, v max(B, B,I). 0, Also we apply throughout the paper the following. THEOREM 1.1 [10--12] Necessary and sufficient conditions for the inequality (1) to be valid on interval (a, b) are: (1) At < cx, when F(a)- F’(a) ...-- F(k)(a)= OandthenC A, F () (b) 0 and then C , Bk. (2) Bk < o, when F(b) F f(b) Our first result is the following simple observation. THEOREM 1.2 Let -x < a < cxz. Necessary and sufficient conditions for the inequality (1) to be validon (a, cxz), when F(cxz) O, is B;(a,),u,v < cxz and then C Bk;(a,),u,v. , Necessity follows from Theorem 1.1. For sufficiency suppose 0 and Bk;(a,),u,v < x. Denote F (k) < x, F(x) f Proof [[F(k)vl[2 and let P(x) Since Jf(x) (Y r(k) x)k-1 f(Y) dy, x > a. Bk;(a,c),u,v < cx, we see, that IF(x)l 1 < --llfvll2 and conclude, that F sufficiency. . (fx x (y x)e(-l)[v(y)[-2 dy )1/2 O, x Applying Theorem 1.1 again, we obtain the WEIGHTED HARDY INEQUALITIES 227 Remark 1.1 Theorem 1.2 implies the limiting case of the assertion (6) 1. The remaining part follows similarly by above, when I1 0, I1 application of the result of [2]. The situation on the left semiaxis is similar, however, for the real line the only limiting cases of (6), that is Io1 0, 0 or 0 valid. are 0, (1, (1, ), fl I11 3 THE CASEk=2 In this section we consider the inequality (3) on (0, oo). The full list of non-trivial cases is the following. Icl + I/1 I1 + I/1 I1 + I/1 Icl + I/1 1. (1.1) F(oo) --0. 2. (2.1) F(0) F(oo) --0, (2.2) V(0) V’(0) 0, (2.3) (2.4) (2.5) 3. (3.1) (3.2) (3.3) (3.4) 4. (4.1) F(oo) F’(oo) 0, V(0)= V’(oo) 0, F’(0) F(oo) 0. F(0) F’(0) V(oo) 0, F(0) F’(0) F’(oo) 0, F(0) F(oo) V’(oo) --0, F’(0) F(oo) F’(oo) 0. F(0) F’(0) F(oo) F’(oo) 0. from The case (1.1) follows from Theorem 1.2, (2.2) and (2.3) Theorem 1.1, (2.4) from [2]. Applying the heuristic principle we show that (1.1) (2.3), (2.1) (3.3), (2.5) (3.4), (3.1) (4.1). 3 need to be treated. Thus, the only four cases of subsection lal + I11 Moreover, we show that the cases (3.2), (3.3) and (3.4) are similar to each other and begin with the first of them. CASE (3.2). It is easy to see, that the characterization problem IIFull2 _< C IIv" ll , F(0)= F’(0)= F’(oo)= 0 M. NASYROVA and V. STEPANOV 228 is equivalent to the following < Cllfvll2, ll(/2f) ull2 Let r 6 f --O. (4) (0, c) be defined by fo Iv1-2 f Iv1-2 (5) and put lul 2 THEOREM 2.1 C (z- X) 2 Iv(x)1-2 dx. The leastpossible constant C in (4) is sandwiched as follows ,, A2;(o,r),u,v + Dr + A1;(r,o),u,(x_r)-lv(x) + B;(r,c),x-r)ux),v. Proof If fo f (6) 0, then for x > r we have ’,-fo (fosts-fo (o )s+f (fots_ fo fo (r (r fX(fs f) y) f (y)dy y) f (y)dy (x r) 1 fx f ds r) f (y)dy. (y (7) For the upper bound we use Theorem 1.1 and Cauchy’s inequality and find r(r y)f(y)dy x X[r’,](x)(x r)u(x) X[r’,](x)u(x) f (y r) f (y)dy 2 (A2;(o,r’),u,v + Dr, h- A1;(r’,o),u,(x-r)-,v(x) q- B;(r’,),(x-r)u(x),v) Ilfvll2. WEIGHTED HARDY INEQUALITIES 229 For the lower bound it is sufficient to arrange a one-to-one isometrical map or between the cones 1 {f L2,v f >_. O, supp f c_ [0, z]} and /22={fL2,v f<0, supp f___[z, cx]) such that (f q- cot f) 0, f (8) We construct cot below, using an idea from ([8], Section 1.8). Now, suppose (4) is tree and the right-hand side of (5) is finite. Since for any function of the form f -+- ogr f, f /21 or o) f + f, f /22 the right-hand side of (7) is nonnegative, we obtain the following inequalities [IX[O,r] (I2f)ul[ 2 < ll(I2(f + rf)) ull2 C II(f + rf)vll2 2C Ilxt0,foll, f c and IIt,a (I2f u[12 2c [[t,afllz, f Then the required lower bound follows from Theorem 1.1. For a possible infinite right hand side of (5) we proceed as before, replacing the weight 0. v(x) by (1 + ex)v(x) and then using limiting guments with To construct the map wr we define for f 6 1 (-i()[ (-, <)i (wrf)(x) where p [0, r] Iv(x)l 2 Iv, ] is given by Iv1-2, Iv s e [0, (s) Then (8) is valid d t, f Theorem 2.1 is proved. [2 t, < f [[2’ f x v, M. NASYROVA and V. STEPANOV 230 Theorem 2.1 has a complete analog for a finite interval, (-cx, 0) or (-cx, cxz) and for an Lp-Lq setting. For a finite interval the Lp-Lq version of Theorem 2.1 follows from the characterization (6) above, which has independently been proved by a different method in [5]. Remark 2.1 CASE (3.4). In this case the characterization of C IIFull2 IIF" l12, F’(0)- F(cx)- F’()= 0 is equivalent to the problem II(J2g) ul12 < C Ilgvll2, g (9) 0, which is dual to (4). Put D 2a,r u 12 (x z’) 2 Iv(x)1-2 dx. THEOREM 2.2 The least possible constant C in (9) is estimated by B2;(r,oo),u,v + Dl,r -+- A1;(o,r),(r-x)u(x),v -+- B1;(O,r),u,(r-x)-lv(x) (10) C Proof This time we use a decomposition for 0 < x < r of the form Jg(x) (y r)g(y) dy ( x) g (r y)g(y) dy and proceed as in the proof of Theorem 2.1. CASE (3.3). Now we have Ilfull2 _< C IIf’ l12, f(0)- F(cz)- F’(oc)= 0, which equivalent to the problem II(Jg) ull < C Ilgvll y g(y) dy O. Making the substitution y g(y) -+ g(y), we make (11) equivalent to (11) WEIGHTED HARDY INEQUALITIES where vl (x) 231 -v(x) and J2g(x) 1 g(y) dy. --f By a known criterion 14] 2211L2,vl,(r,c)-+ L2,u,(r,c) L2,v,(r,ee) L2,u,(r,) Thus, using the decomposition x -2g(x) dy (y- "c)g(y) y z-x fr r. foX Xfxr g dy y)g(y), 0 < x < r y (3 and arguing as in the proof of Theorems 2.1 and 2.2 we obtain THEOREM 2.3 C The least constant C in (11) is estimated by B2;(r,),u,v -+- z"-1 (D2,r -t- A1;(O,r),(r-x)u(x),x-lv(x) -’l-Bl.(O,r),xu(x),(r-x)-Iv(x)) (12) where 2 D2,r f0 x2lu(x)l 2 dx (x z-)2lv(x)1-2 dx CASE (3.1). Now we have equivalence of IlFull2 C [[F"v[[ 2 F(0) F’(0) F(cx) 0 and y II(lzf) ullz _< C Ilfvll2, f0c(f0 f) dy where the integral is interpreted in the Riemann sense. We need the following simple note. O, (13) M. NASYROVA and V. STEPANOV 232 A locally integrable function f satisfies LEMMA fo(foY f) (14) 0 dy (0, cxz) such, that if and only if there exist ) z y foZf--O fo (fZf)dy-- fz (fz f) and dy. (15) Proof Suppose (14) be valid, then the first assertion in (15) is trivial. Then we have O-- ji (foYf) dY-- foL (foYf) -ji’ (fy’f) dy-k- dy-t- fc (foYf) f (fYf) dy- dy. Conversing the last line, we prove the "if" part. Suppose ) 6 (0, c) and put f0 vz THEOREM 2.4 C 101-2 101-2. The least possible constant C in (13) is given by A2;(0,c),u,v + sup (Al’(rz,Z),u,(x-rz)-’v(x) + Bl’(rz,.),(x-r)u(x),v). )>0 Proof We begin with the upper bound. Let f Hence, (15) is true for some ) I2 f (x) fo (;k 6 L2,v and (14) be valid. (0, cx). Then for x > y) f (y) dy + (x . (16) we have y) f (y) dy. Using this and applying Theorems 1.1 and 2.1, we obtain WEIGHTED HARDY INEQUALITIES sup A2;(O,r),u,v + x)21v(x)1-2 dx+ (’cz lu 233 A1; (rx,Z),u, (x-rx)-l v(x) -t- lul 2 (z x)21v(x)1-2 dx + Aa;(x,,, - f0 For the lower bound we suppose ; [rz, )] by Pl [0, rz] Given f L2,v, supp outside [:, )] and Iv1-2- l(s) 6 Iv1-2 f c_ [0, rz], f(x) f (x) (0, x) and define the function s[0 > 0 we define fl(x) to be zero x Iv(x)l (rx, .). Then, by the argument from the proof of Theorem 2.1 we have f+ Now, given ) (0, cx) we find IX f0 x21v(x)1-2 and define the function/92 fo x2lv(x)1-2 dx fl IX()) (Ix 0. (17) (0, cx) from equation x)21v(x)1-2 dx [0, )] -+ [), Ix] by dx (Ix x)2lv(x)1-2 dx. (18) M. NASYROVA and V. STEPANOV 234 Put f+fl f0 and define fz(x) O, x t [k, tt] and /2(x) -1 /92 (19) (x)Iv(x) 12 Then Now, put fo+ f2. f Then (17) implies Let us show that ) fc (fy ) dy dy. Indeed, applying (18) and (19) we find cx Y f dy (lZ Y) f2(Y) dy (lZ sfo(s)ds=foZ(fyZf) pz(s)) f2(P2(s)) dp2(s) dy. 37 The lemma implies that satisfies (14) and, consequently, admissible for (13). Since functions f0 form a cone used for the lower bound in the proof of Theorem 2.1, we obtain C sup (Aa;(o,rx),u,v + Dr + Al.(rx,z.),u,(x_vx)-lv(x) -t-- Bl.(rx,z.),(x-rx)u(x),v) >> )>o sup (A1;(rx,)),u,(x-rx)-v(x) + Bl’(rx,.),(x-rDu(x),v). >> )>0 WEIGHTED HARDY INEQUALITIES 235 For the remaining part of lower bound we suppose, that 0 < and choose/z (), a) such that (0" x)21v(x)1-2 dx x)21v(x)1-2 dx. (a - - co. Let f L2,v, supp co, whereas a Obviously,/z f(x) >_ O. Define the function p [;,/z] -+ [/z, a] such that (a x )2 Iv(x)l -2 dx )v < a < co (a f [;k,/z], x)2lv(x)1-2 dx (s) and define fl (x) 0, x [/x, a] and = fl (X) (/Z, o’). X (a -p-l(x))lv(x)l 2 Put Then it is routine to verify that Ilfvll 2llfvll22 and fo(foYf) dy-O. Applying Theorem 1.1 we find - C >> A2;(;,z),u,v, 0< . < lZ < co. co and then ;k --+ 0, we obtain the remaining part of the lower Letting a bound. Theorem 2.4 is proved. CASE (2.5). We have Fu 112 _< c F"v 2, F’(0) F(co) 0. (20) M. NASYROVA and V. STEPANOV 236 THEOREM 2.5 by (10). Proof The estimate for the least constant C in (20) is established Since {F" F’(0) 0} F(c) {F" F’(0) F(cx) F’(cx) 0}, we find by Theorem 2.2, that C >> B2;(r,oo),u,v + Dl,r + A1;(o,r),(r-x)u(x),v + B1;(o,r),u,(r-x)-lv(x). Now, suppose B2;(r,oo),u,v + Dl,r + Al’(O,r),(r-x)u(x),v + Bl’(O,r),u,(r-x)-v(x) < (x) and, consequently, (x-s Iv(x)l dx < oo, s > r. Hence, the function l(s) (x s)F"(x) dx is defined for all s > 0,/ (cxa) 0 and P" by Theorem 2.2 we obtain the upper bound. F". It implies that P F and CASE (2.1). Now we have IIFull2 C [[F"v[[ 2 F(0)= F(oo)- 0. (21) Analogously to the case (2.5) we prove the following. THEOREM 2.6 The estimate for the least constant C in (21) is established by (12). CASE (4.1). Now the problem is to characterize Ilgull2 c liE"vii2, F(0)= F’(0)= F(cx)- F’(c)= 0. (22) WEIGHTED HARDY INEQUALITIES 237 The estimate for the least constant C in (22) is established The upper bound immediately follows from Theorem 2.4. For the lower bound suppose (22) is valid. Then it is valid for the weight re(x) (1 + ex)lv(x)[ instead of v(x). It is easy to see that Proof {F" IlF’Zvell2 < cx3, {F" F(0) IIF’vll2 < F’(0) F(cxz)- F’(c)- 0} , F(0)= F’(0)= F(cx)- 0}. Consequently, by Theorem 2.4 we have C >> A2;(a,),u,v + sup (Al.(zx,)O,u,(x-rz)-,v(x) + Bl’(rx,)),(x-rz)u(x),v). )>0 The Fatou theorem brings the required lower bound. Remark 2.2 The characterization of cases (2.1), (2.5) and (4.1) on a finite interval is different (for (2.1) and (2.5) see [6]) and (4.1) is so far unknown. Acknowledgements The authors wish to thank Professors Hans Triebel and Gordon Sinnamon for valuable discussion of the paper. 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